Highly Undecidable Problems about Recognizability by Tiling Systems

摘要

Altenbernd, Thomas and Wohrle have considered acceptance of languages of infinite two-dimensional words (infinite pictures) by finite tiling systems, with usual acceptance conditions, such as the Buchi and Muller ones, in [1]. It was proved in [9] that it is undecidable whether a Biichi-recognizable language of infinite pictures is E-recognizable (respectively, A-recognizable). We show here that these two decision problems are actually ∏_2~1-complete, hence located at the second level of the analytical hierarchy, and "highly undecidable". We give the exact degree of numerous other undecidable problems for Biichi-recognizable languages of infinite pictures. In particular, the non-emptiness and the infiniteness problems are ∑_1~1-complete, and the universality problem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, are all ∏_2~1-complete. It is also ∏_2~1-complete to determine whether a given Buechi recognizable language of infinite pictures can be accepted row by row using an automaton model over ordinal words of length ω~2.